Several (partial) results are available from the technical literature
about the well-posedness of the MHD model in the strong and weak
formulation, e.g., problems ()
and () and their variants, cf.
[1]}, [2]}, [3]}, [4]}, [5]}, [6]}
and the citations therein.
Some of these results concerning existence, uniqueness and stability
of the solution fields \((\mathbf {u},p,\mathbf {E},\mathbf {B})\)  have been derived under
specific assumptions that may reduce the generality of the model.
To the best of our knowledge, the full mathematical understanding of
the MHD model is still an open issue and an active research area.
This topic is beyond the scope of our work; thus, we shall simply
assume that the MHD model is well-posed, at least in the setting that
we are using in this paper.
